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What is the difference between direct sum $\bigoplus_{i\in I} M_i$ and restricted product $\prod'_{i \in I}M_i$ of abelian group?

There are many types of products you can do. This only applies to products of infinitely many (normal) subgroups, but also not just to abelian groups. There are different names for these depending on ...
David A. Craven's user avatar
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External direct sum $U_1\oplus U_2$ isomorphic to $U_1+U_2$ does not necessarily imply that $U_1\cap U_2 = \{0\}$

There are two related but distinct questions here: Is the map $U_1 \oplus U_2 \to V$ given by $(u_1, u_2) \mapsto u_1 + u_2$ an isomorphism onto $U_1 + U_2$? Equivalently, is this map injective (...
ronno's user avatar
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External direct sum $U_1\oplus U_2$ isomorphic to $U_1+U_2$ does not necessarily imply that $U_1\cap U_2 = \{0\}$

[I take it that $U_1\oplus U_2$ refers to the external direct sum.] Let $V$ be the vector space of real sequences, $\{(v_n)_{n=0}^{\infty}\}$. Let $U_1:= \{v\in V\mid v_{2i+1}=0 \text{ for all}\ i\...
ancient mathematician's user avatar

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